How can I rebuilt a similar time series with less frequencies after performing an fft?

Hello there,

I've been facing a problem for some time and I hope someone can help me on that.

I am using the fft on a signal (water waves elevation) in order to get its corresponding spectrum. As I want to simulate a similar signal in the time domain, I need to know the frequencies and their corresponding amplitude. Besides, each of the frequency is also needed for computing a wave force related to it and applying on a body.

I would like to "cut" the frequency range I take into consideration as after a certain value their amplitudes are so small that they are irrelevant for the force calculation and do not affect much the water wave elevation signal.

I tried to do it for various number of frequencies, but every time I have the new time series and that I perform an fft on it, I always have lower amplitudes than the original ones, and for this reason I simply can not use it.

I join hereinafter an example of what I am doing:

    fq = 100; %Hz (Sampling time 0.01)

    L1 = length(xx2); % xx2 is the original waves elevation signal

    NFFT = 2^nextpow2(L1);

    Y1 = fft(xx2,NFFT); %Fourier transform of the signal

    Y11 = abs(Y1(1:NFFT/2+1)); %Selection of the left-hand side of the fft

    f = fq/2*linspace(0,1,NFFT/2+1);  Frequency vector correponding to Y11 [Hz]

    wi = 2*pi()*f;      %Conversion of f from Hz to rad.s-1

    Y_norm = Y11/L1;    %Normalisation of the experimental time serie FFT;

    figure;plot(wi,Y_norm);

    wn = 0:0.04:10.22;   %Vector of desired frequencies

     m3 = zeros(size(wn));

     p0=0;               
     for k = 1:length(wn)
     p = find(wi>=wn(k),1);
     m3(k) = sum(Y_norm(p0+1:p));
     p0 = p;
     end

In this last part I try to take into account the frequencies from the fft which are smaller than the ones I'm aiming at to get the corresponding frequency.

     t = 0:0.01:200;
     yt = zeros(length(t),1);
     teta = 2*pi*rand(size(m3));
     for j = 1:length(t)
     yt(j) = sum(m3.*cos(wn*t(j)+teta) );
     end

This is for recomposing a similar time series with new phases. My problem is that everytime I try to apply an fft to the "yt" signal, I obtain a spectrum with more or less the same shape, but lower amplitudes than in the original one, thus preventing me to go further...

I thank you for any help or suggestion you can give me.

Matthieu

[PT]

Wouldn't filtering the signal have the same effect?

Answer 1

[Youssef Khmou]

hi, try this modified version :

    % ORIGINAL SIGNAL
    fq = 100; %Hz (Sampling time 0.01)
    t = 0:1/fq:200;
    wn = 0:0.04:10.22;   %Vector of desired frequencies
    xx2=0;
    for n=1:length(wn)
        xx2=xx2+cos(2*pi*t*wn(n));
    end
    xx2=xx2/length(wn);
    figure, plot(t,xx2);
    AX=axis;

    % FORWARD DISCRET FOURIER TRANSFORM
    L=length(xx2);
    N=ceil(log2(L));
    fxx2=fft(xx2,2^N)/L/2;
    f=(fq/2^N)*(0:2^(N-1)-1);
    FF=abs(fxx2(1:end/2));
    figure, plot(f,FF)

    %  RECONSTRUCTION   :  NOT FINISHED YET
     wn = 0:0.04:10.22;   %Vector of desired frequencies
     m3 = zeros(size(wn));
     p0=0;               
     for k = 1:length(wn)
     p = find(f>=wn(k),1);
     m3(k) = sum(FF(p0+1:p));
     p0 = p;
     end

     t = 0:0.01:200;
     yt = zeros(length(t),1);
     teta = 2*pi*randn(size(m3));
     for j = 1:length(t)
     yt(j) = sum(m3.*cos(wn*t(j)+teta) );
     end
     figure, plot(t,yt); axis(AX);

       % FORWARD DISCRET FOURIER TRANSFORM of the RECONSTRUCTED SIGNAL

    fyt=fft(yt,2^N)/L/2;
    f=(fq/2^N)*(0:2^(N-1)-1);
    FY=abs(fyt(1:end/2));
    figure, plot(f,FY)

We discuss it , next....